IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______
Color Energy of Generalised Complements of a Graph Swati Nayak, Sabitha D’Souza∗, Gowtham H. J. and Pradeep G. Bhat,
Abstract—The color energy of a graph is defined as sum A coloring of graph G is a coloring of its vertices such of absolute color eigenvalues of graph, denoted by Ec(G). Let that no two adjacent vertices receive the same color. The Gc = (V,E) be a color graph and P = {V1,V2,...,Vk} be a minimum number of colors needed for coloring a graph G partition of V of order k ≥ 1. The k -color complement {G }P c k is called chromatic number, denoted by χ(G). of Gc is defined as follows: For all Vi and Vj in P , i 6= j, remove the edges between Vi The color matrix is as follows. If c(vi) is the color of vi, and Vj and add the edges which are not in Gc such that end then 1, if v ∼ v with c(v ) 6= c(v ), vertices have different colors. For each set Vr in the partition i j i j P , remove the edges of Gc inside Vr, and add the edges of Gc aij = −1, if vi vj with c(vi) = c(vj), (the complement of Gc ) joining the vertices of Vr. The graph P 0, otherwise {Gc}k(i) thus obtained is called the k(i)− color complement The matrix thus obtained is the L-matrix of the colored G P V of c with respect to the partition of . In this paper we G A (G) characterize generalized color complements of some graphs. We graph denoted by c . Color energy of colored graph also compute color energy of generalised complements of star, is the sum of the absolute colored eigenvalues. complete, complete bipartite, crown, cocktail party, double star i.e., n and friendship graphs. X Ec(G) = λi. Index Terms—k−color complement, k(i)−color complement, i=1 color energy, color spectrum. In 2013, the concept of color energy of a graph was I.INTRODUCTION introduced by Chandrashekar Adiga et.al [12]. Let G = (V,E) be a colored graph. Then the complement Let G = (V,E) be a graph of order n. The complement of colored graph G denoted by G has the same coloring of of a graph G, denoted as G has the same vertex set as that c G with the following properties: of G, but two vertices are adjacent in G if and only if they are not adjacent in G. If G is isomorphic to G then G is • vi and vj are adjacent in Gc, if vi and vj are non- said to be self-complementary graph. For all notations and adjacent in G with c(vi) 6= c(vj). terminologies we refer [1], [2]. • vi and vj are non-adjacent in Gc, if vi and vj are non- The concept of energy of a graph was introduced by I. adjacent in G with c(vi) = c(vj). Gutman in the year 1978 as the sum of absolute eigenvalues • vi and vj are non-adjacent in Gc, if vi and vj are of graph G. More on energy of graphs we refer to [3], [4], adjacent in G. [5], [6], [7], [8], [9], [13], [14], [15]. In 1998 E. Sampathku- The paper is organised as follows. In section II, we mar et.al defined generalised complements of a graph i.e, introduce the concept of generalised color complements of P k−complement Gk of a graph G and k(i)−complement a graph. The preliminary results are shown in section III. P The characterisation of generalised color complements is Gk(i) of a graph G respectively [10],[11]. Let G = (V,E) be a graph and P = {V1,V2,...,Vk} be presented in section IV. The color energy of generalised P complements of few families of graph is obtained in section a partition of V of order k ≥ 1. The k− complement Gk of G [10] is defined as follows: For all Vi and Vj in P , i 6= j, V. remove the edges between Vi and Vj and add the edges which are not in G. The graph G is k− self complementary (k − II.GENERALISEDCOLORCOMPLEMENTSOFAGRAPH s.c) with respect to P if GP =∼ G. Further, G is k−co−self k In this section we introduce a concept of generalised color complementary (k − co − s.c.) if GP =∼ G. k complements of a graph. For each set Vr in the partition P , remove the edges of G inside Vr, and add the edges of G joining the ver- Definition 1. Let Gc = (V,E) be a color graph and P = P tices of Vr. The graph Gk(i) thus obtained is called the {V1,V2,...,Vk} be a partition of V of order k ≥ 1. P k(i)−complement [11] of G with respect to the partition P The k -color complement {Gc}k of Gc is defined as follows: of V . The graph G is k(i)−self complementary (k(i) − s.c) For all Vi and Vj in P , i 6= j, remove the edges between P ∼ if Gk(i) = G for some partition P of order k. Further, G is Vi and Vj and add the edges which are not in Gc such that P ∼ end vertices have different colors. k(i)−co−self complementary (k(i) − co − s.c.) if Gk = G.
Manuscript received March 10, 2020; revised July 10, 2020. • The graph Gc is k− self color complementary (k−s.c.c) P ∼ Department of Mathematics, Manipal Institute of Technology, with respect to P if {Gc}k = Gc. Manipal Academy of Higher Education, Manipal, 576104 India • Further, Gc is k−co−self color complementary (k − e-mail: ([email protected], [email protected], P ∼ [email protected], [email protected]). co − s.c.c) if {Gc}k = Gc. ∗ Corresponding author: SABITHA D’SOUZA, Department of Math- ematics, Manipal Institute of Technology, Manipal Academy of Higher Example 2. Let {V1,V2} be two partition of vertex set of Education, Manipal, 576104 India Cycle graph C4.
Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020)
IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______
Proposition 8. [5] Let M,N,P,Q be matrices, and let M C C C C 1 2 1 2 be invertible.
MN Let S= PQ Then det S = det M. det[Q − PM −1N]. If M and P C C 2 1 commute, then det S = det [MQ − PN]. C2 C1 P ()C ()C 4 c { 4 c} 2 IV. CHARACTERISATIONOFGENERALISEDCOLOR Fig. 1. Colored cycle C4 and its 2− complement. COMPLEMENTSOFGRAPH
Definition 3. For each set V in the partition P , remove r The following Proposition highlights the isomorphism the edges of G inside V , and add the edges of G (the c r c between the generalised color complements of a graph. complement of Gc ) joining the vertices of Vr. The graph P {Gc}k(i) thus obtained is called the k(i)− color complement Proposition 9. For any Gc, of Gc with respect to the partition P of V . P ∼ P 1) (Gc)k = (Gc)k . • G k(i)− (k(i)− The graph c is self color complementary P ∼ P P ∼ 2) (Gc)k(i) = (Gc)k(i). s.c.c) if {Gc}k(i) = Gc for some partition P of order k. Proof: • Further, Gc is k(i)−co−self color complementary P ∼ 1) Consider two vertices u and v in G . Then u ∼ v in (k(i) − co − s.c.c) if {Gc} = Gc. c k(i) P P (Gc)k ↔ u v in (Gc)k . Example 4. ↔ u and v are in the same set in P , and are non-adjacent in Gc, or they are in different sets in P , and u ∼ v in Gc if c(u) 6= c(v). ↔ u and v are in same set in P , and u ∼ v in Gc C 1 C C C if c(u) 6= c(v), or they are in different sets in P , and 2 1 2 u v in Gc. P ↔ u ∼ v in (Gc)k if c(u) 6= c(v). P 2) Let u ∼ v in (Gc)k(i). P ↔ u v in (Gc)k(i). ↔ u and v are in same set in P , and u ∼ v in Gc C 2 C C C if c(u) 6= c(v), or they are in different sets in P , and 3 2 3 G { p u v in Gc. c Gc} 2 (i) ↔ u and v are in same set in P , u v in Gc, or they are in different sets in P , and u ∼ v in Gc if c(u) 6= c(v) . P P ↔ u ∼ v in (Gc)k(i) if c(u) 6= c(v). Fig. 2. Graph Gc and {Gc}2(i).
Definition 5. In a generalised color complement of graph As a consequence of Proposition 9, we have G, the indegree of vertex v denoted by i (v) is defined as d Corollary 10. For any graph G , number of edges incident to the vertex v inside the partite. c Outdegree of vertex v denoted by od(v) is defined as number P ∼ P ∼ 1) (Gc)k = Gc if and only if (Gc)k = Gc. of edges incident to the vertex v outside the partite. P ∼ P ∼ 2) (Gc)k(i) = Gc if and only if (Gc)k(i) = Gc.
III.PRELIMINARIES P ∼ P Proposition 11. 1) (Gc)k = (Gc)k(i). Now we present few results on k and k(i) self complemen- P ∼ P tary graphs which are extensively used to prove our main 2) (Gc)k(i) = (Gc)k . results. Corollary 12. For any graph Gc, Proposition 6. [10] The k−complement and k(i)− comple- 1) (G )P ∼ (G )P ∼ (G )P . ment of G are related as follows: c k = c k = c k(i) P ∼ P 2) (G )P =∼ (G )P =∼ (G )P . 1) Gk = Gk(i). c k(i) c k(i) c k P ∼ P 2) Gk(i) = Gk . P ∼ P ∼ Corollary 13. 1) (Gc)k = Gc ↔ (Gc)k(i) = Gc. A0 A1 P ∼ P ∼ Proposition 7. [5] Let A= be a symmetric 2 × 2 2) (Gc)k(i) = Gc ↔ (Gc)k = Gc. A1 A0 block matrix. Then the spectrum of A is the union of the Example 14. Let {V1,V2} be two partition of the vertex set spectra of A0 + A1 and A0 − A1. of the graph (P4)c.
Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020)
IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______
P 0 PiPj − q edges. So C1 C C C c ()P : 2 1 2 i 1 0 ≥ [(k − 1)(2p − k) − 2qc]. Fig. 3. Colored path P4 and its 2−complement. 4 ∼ P Since (P4)c = {(P4)c}2 with respect to the partition Corollary 16. If Gc(p, q) is k(i)−co−s.c.c., then 1 1 {V1,V2}, (P4)c is 2− self color complementary graph. [(k−1)(2p−k)−2q0 ] ≤ q ≤ [2p(p−k)+k(k−1)+2q0 ]. 4 c 4 c Remark 17. 1) Complete bipartite graph Kl,m of order n, 0 Star graph K1,n−1 and Crown graph Sn are not k−self color and k(i)−co−self color complementary for any _ n. 2) Cycle graph C , n ≤ 10, is 2−self color and 2(i) − C1 C C C n ()P : 2 1 2 co−self color complementary for n = 7, 8. 4 c 3) Path graphs Pn, n ≤ 10, is 2-self color and 2(i) − co−self color complementary except for n = 2, 3, 10. 4) Every Complete bipartite graph Kl,m is k(i)−self color and k−co−self color complementary with respect to the P C C C C {()P : 1 2 1 2 partition of same color class. 4 c } 5) All even Cycle graph Cn and Path graph Pn are 2(i) 2(i)−self color and 2−co−self color complementary for n ≤ 10. Fig. 4. Complement and 2(i)−complement of colored path P4. Theorem 18. (W1,n)c is not k− self color complementary except for n = 7 and k = 3. ∼ P Since (P4)c = {(P4)c}2(i) with respect to the partition Proof: Let (W1,n)c be color wheel graph with O as apex {V1,V2}, (P4)c is 2(i)−co−self color complementary graph. vertex and F = {fi/i = 1, 2, . . . , n} as set of peripheral Theorem 15. If a Gc(p, q) is k − s.c.c, then vertices. Let P = {V1,V2,...,Vk} be a partition of vertex O 1 0 1 0 set of wheel. We observe that the apex vertex remains [(k−1)(2p−k)−2qc] ≤ q ≤ [2p(p−k)+k(k−1)+2qc] P 4 4 either as apex vertex or peripheral vertex in {(W1,n)c}k . But where q0 is number of pairs of non- adjacent vertices with P c O cannot be apex vertex in {(W1,n)c}k . Suppose O belongs same color between the partites. to the set V1. Then, there should be at least 3 peripheral vertices in V so that deg(O) = 3 and n − 3 peripheral Proof: |V | = P , 1 ≤ i ≤ k q0 1 Let i i . Let c be the number vertices must be in other sets of P . Let u, v and w be of pair of non-adjacent vertices of same color between the peripheral vertices in V1 so that u ∼ v, w and v w. Then partites in graph Gc. Then total number of edges between P id(O) = id(u) = 3 and id(v) = id(w) = 2. Vi and Vj in P , i 6= j in both Gc and (Gc)k is X = P 0 Let us take following two cases. Let us assign numbers PiPj − qc. i Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ n + 3 n + 1 Hence deg(u) = , deg(v) = deg(w) = and Hence (W1,n)c is 2(i)−self color complementary if and only 2 2 deg(O) = 3. if n = 8 with respect to partition {V1,V2} such that V1 has apex vertex and V has remaining vertices. Subcase 2: Let V1 = {(1,O), (3, v), (2, u), (4, w)}. 2 n − 3 n − 5 Then o (u) = , o (v) = n − 4, o (w) = and d 2 d d 2 od(O) = 0. Theorem 21. (W1,n)c is 2− co− self color complementary n − 3 if and only if n = 8. Therefore deg(u) = , deg(v) = n − 2, deg(w) = 2 n − 1 Proof: Proof follows from Theorem 20 and Corollary and deg(O) = 3. 2 13. These two subcases are discarded as the criteria fails. P P 0 Subcase 3: Let V1 = {(1,O), (2, v), (4, u), (3, w)}. Remark 22. If Gk has q edges and (Gc)k has q edges, n − 5 then q = q0 + q0 with respect to same partition, where q0 is Then od(u) = n − 3, od(v) = od(w) = and od(O) = c c 2 the number of pair of vertices of same color class. 0. n − 1 Thus deg(u) = n, deg(v) = deg(w) = and deg(O) = 2 3. V. COLORENERGYOFGENERALISEDCOMPLEMENTSOF Therefore, u is a apex vertex and v, w and O are peripheral SOME STANDARD GRAPHS vertices for n = 7. i.e, deg(v) = deg(w) = 3 for n = 7. Possible partitions are : In this section we consider the colored graph with respect {{O, u, v, w}, {x, y, z, m}} to minimum number of colors. We determine the color {{O, u, v, w}, {y, z, m}, {x}} energy and color spectrum of generalised complements of {{O, u, v, w}, {x, y, z}, {m}} some families of graphs. {{O, u, v, w}, {x, y, m}, {z}} Let G = (V,E) be a graph. Consider the coloring of G {{O, u, v, w}, {x, z, m}, {y}} with minimum number of colors, i.e, χ(G) colors. In such a {{O, u, v, w}, {x, y}, {z, m}} case, we denote Ac(G) by Aχ(G) and Ec(G) by Eχ(G). {{O, u, v, w}, {x, z}, {y, m}} {{O, u, v, w}, {x, m}, {y, z}} Theorem 23. Let {K1,n−1}c be colored star graph with par- tition P = {V1,V2,...,Vk} where V1 = {v1, v2, . . . , vm}, Out of these eight partitions, we see that (W1,7)c is isomor- P v1 be central vertex. Then characteristic polynomial of k− phic to {(W1,7)c}k for {{O, u, v, w}, {x, y}, {z, m}} only. n−3 3 2 complement of {K1,n−1}c is (λ − 1) [λ + (n − 3)λ − (n + m − 3)λ − (m − 1)(n − m − 1)]. (2,v) (3,x) b (2,v) (3,x) b b b b (4,u) Proof: Let {K1,n−1}c be a colored star graph on n b (4,u) b vertices. Since χ({K1,n−1}c) = 2 , adjacency matrix of k− (2,y) b b (1, O) (1, O) b (3,w) color complement of star graph is (2,y) b b (3,w) P Aχ({K1,n−1}c)k = b b b b (2,m) 0 J1×m−1 O1×n−m (2,m) (3,z) (3,z) Jm−1×1 −Bm−1 −Jm−1×n−m , On−m×1 −Jn−m×m−1 −Bn−m Fig. 5. Color wheel graph (W1,7)c and its 3-color complement n×n P where O is the matrix of all zero’s, J is the matrix of all {(W1,7)c}3 10s and B is the adjacency matrix of complete subgraph. |λI − A ({K } )P | = Hence (W1,n)c is not k− self color complement except χ 1,n−1 c k λ −J1×m−1 O1×n−m (W1,7)c for k = 3. − Jm−1×1 (λI + B)m−1 Jm−1×n−m Corollary 19. (W1,7)c is 3(i)− co− self color complemen- On−m×1 Jn−m×m−1 (λI + B)n−m tary graph. Step 1: For rows i = 3, 4, . . . , n, by performing row n−3 operation Ri −→ Ri − Ri−1, we get (λ − 1) det(C). Proof: Proof follows from Theorem 18 and Corollary 0 13. Step 2: In det(C), by performing Ci = Ci +Ci+1 +...+Cn, i = 2, 3, . . . , n − 1, det(C) reduces to order 3. Theorem 20. (W ) is 2(i)−self color complementary if 1,n c Thus and only if n = 8. λ −(m − 1) 0 Proof: Let us take the following two cases. det(C) = −1 λ + n − 2 n − m . Case 1: Let P = {V1,V2} be two partition of (W1,n)c such 1 0 λ − 1 that < V1 >= O and < V2 >= {fi/i = 1, 2, . . . , n}, where Step 3: By simplifying, we get fi denotes the peripheral vertex. Then the degree of apex P n−3 3 2 P |λI −Aχ({K1,n−1}c) | = (λ−1) [λ +(n−3)λ −(n+ vertex in {(W1,n)c}2(i) will be n as all the edges between k P m − 3)λ − (m − 1)(n − m − 1)]. V1 and V2 remain unaltered so that {(W1,n)c}2(i) = K1 + (Cn)c. This is possible only when n = 8 as C8 is self color Theorem 24. Let {K1,n−1}c be colored star graph with par- complementary. tition P = {V1,V2,...,Vk} where V1 = {v1, v2, . . . , vm}, Case 2: Let V1 be the set of apex and some peripheral v1 being central vertex. Then characteristic polynomial of n−3 3 vertices and V2 be the set of remaining vertices. Then k(i)− complement of {K1,n−1}c is (λ − 1) [λ + (n − P 2 {(W1,n)c}2(i) results into disconnected graphs. 3)λ − (2n − m − 2)λ − (m − 2)(n − m)]. Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ P Proof: Adjacency matrix of ({K1,n−1}c)k(i) is Step 1: Applying the row operations Ri → Ri −Ri+1, where P i 6= a, b, c, d result in new determinant (λ − 1)n−4 det(C). Aχ({K1,n−1}c)k(i) = Step 2: On applying column operations 0 O1×m−1 J1×n−m Ci → Ci + Ci−1 + ... + C1, where i = a, a − 1,..., 2, Om−1×1 −Bm−1 −Jm−1×n−m J −J −B Cj → Cj +Cj−1 +...+Ca+1, where j = b, b−1, . . . , a+2, n−m×1 n−m×m−1 n−m n×n Consider Cr → Cr +Cr−1 +...+Cb+1, where r = c, c−1, . . . , b+ 2 P and Cs → Cs + Cs−1 + ... + Cc+1, s = d, d − 1, . . . , c + 2 |λI − Aχ({K1,n−1}c) | = k(i) in det(C), we get a new determinant say det(D). λ O −J 1×m−1 1×n−m Step 3: Expanding det(D) along the rows except at O (λI + B) J m−1×1 m−1 m−1×n−m ath, bth, cth and dth rows result into det(E), which is given − J J (λI + B) n−m×1 n−m×m−1 n−m by Repeating the step 1 and step 2 of Theorem 23, we obtain λ + a − 1 −b c 0 the result. −a λ + b − 1 0 d det(E) = a 0 λ + c − 1 −d Theorem 25. If {K1,n−1}c is colored star graph with k 0 b −c λ + d − 1 Step 4: From expansion of det(E) and by back substitution, partition P = {V1,V2,...,Vk}, where V1 consists only central vertex, then we obtain the result. P 1) Eχ({K1,n−1}c)k = 2(n − 2). Corollary 28. Characteristic polynomial of 2 and P 2(i)−color complement of {K } with respect to partition 2) Eχ({K1,n−1}c)k(i) = 2(n − 1). l,m c of same color classes are [λ+l −1][λ+m−1](λ−1)l+m−2 Proof: Since {K1,n−1}c is k−co self and k(i)− self and [λ + l + m − 1](λ − 1)l+m−1 respectively and the color complement with respect to the above partition, spec- respective color energies are E ({K } )P = 2(l +m−2) P χ l,m c k trum and enegry of ({K1,n−1}c)k and and E ({K } )P = 2(l + m − 1). P χ l,m c k(i) ({K1,n−1}c)k(i) respectively are 0 −(n − 2) 1 Proof: Since {Kl,m}c with respect to the above partition Spec ({K } )P = and χ 1,n−1 c k 1 1 n − 2 is 2−co−self and 2(i)−self color complementary, proof P follows from theorems [3.10, 3.11] in [12]. Eχ({K1,n−1}c) = 2(n − 2). k P −(n − 1) 1 Specχ({K1,n−1}c) = and k(i) 1 n − 1 Corollary 29. Characteristic polynomial of 2 and 2(i)− P Eχ({K1,n−1}c)k(i) = 2(n − 1). color complement of {Kl,m}c with partition P = {V1,V2}, Proof is similar to the theorems [3.8, 3.9] of [12]. where |V1| = 1 is 1) (λ − 1)l+m−3[λ3 + (l + m − 3)λ2 − (2l + m − 3)λ + Observation 26. 1) E ({K } )P = E({K })P . χ n c k n k (l + m − lm − 1)]. 2) E ({K } )P = E({K })P . χ n c k(i) n k(i) 2) (λ − 1)l+m−3[λ3 + (l + m − 3)λ2 − {l(l − 2) − 3(m − Theorem 27. Let P = {V1,V2} be a partition of colored 1)}λ + {m(3 − 2l) + (l − 1)}]. complete bipartite graph {K } such that hV ∪ V i is l,m c 1 2 Theorem 30. Let (S0) be colored crown graph with vertex union of color complete bipartite subgraphs. Then char- n c set V = {u , u , . . . , u , v , v , . . . , v } and partition acteristic polynomial of 2 and 2(i)− color complement of 1 2 n 1 2 n P = {V ,V }, where V = {u v /1 ≤ i ≤ r, |u | = a, |v | = {K } is (λ − 1)n−4[λ4 + λ3(n − 4) + λ2(ad − 3n + bc + 1 2 1 i i i i l,m c b} and V = {u v /i = r+1 to n, |u | = n−a, |v | = n−b}. 6) + λ(3n − 2ad − 2bc − abd − abc − acd − bcd − 4) + (ad − 2 i i i i Then n + bc + abc + abd + acd + bcd − 3abcd + 1)]. 0 P p 2 1) Eχ({Sn}c)2 = 2[a(n−a)+n−2+ n − 3a(n − a)]. 0 P p 2 Proof: 2 and 2(i)− color complements of complete bi- 2) Eχ{(Sn)c}2(i) = 3n − 4 − 2a + n + 12a(n − a). partite graph is union of colored complete bipartite subgraph Proof: 1) Adjacency matrix of 2−color complement of i.e, {K } ∪ {K } . a,b c c,d c colored crown graph is A ({S0} )P = Since the chromatic number of complete bipartite graph χ n c 2 is 2, adjacency matrix of 2−color complement of complete −Ba −Ja×n−a Ba×b Oa×n−b P A ({K } ) = − Jn−a×a −Bn−a On−a×b Bn−a×n−b bipartite graph is χ l,m c 2 , B O −B −J b×a b×n−a b b×n−b −Ba Ja×b −Ja×c 0a×d On−b×a Bn−b×n−a −Jn−b×b −Bn−b Jb×a −Bb 0b×c −Jb×d , where B is the adjacency matrix of complete subgraph and −Jc×a 0c×b −Bc Jc×d O 0 −J J −B is the matrix of all zeroes. d×a d×b d×c d n×n 0 P |λI − Aχ({Sn}c)2 | = 0 where J is the matrix of all 1 s, B is the adjacency matrix (λI + B) J −B O of complete subgraph and 0 is the matrix of all zeroes. J (λI + B) O −B P |λI − Aχ({Kl,m}c)2 | = − B O (λI + B) J O −B J (λI + B) (λI + B)a −Ja×b Ja×c 0a×d −Jb×a (λI + B)b 0b×c Jb×d Jc×a 0c×b (λI + B)c −Jc×d P Q Above determinant is of the form . 0d×a Jd×b −Jd×c (λI + B)d Q P Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ Hence, spectrum is the union of spectra of |P + Q| and Theorem 32. Let (Kn×2)c be a colored cocktail party graph |P − Q|. with partition P = {V ,V ,...,V } of same color class. 1 2 k λI J Then |P + Q| = . J λI P n×n 1) Eχ{(Kn×2)c}k = 2n. (λ − 2)I + 2J J 2) E {(K ) }P = 6(n − 1). |P − Q| = . χ n×2 c k(i) J (λ − 2)I + 2J n×n Proof: Colored cocktail party graph is k−co-self and Applying the row and column operations respectively R → i k(i)− self color complementary with respect to partition of R − R , i = 1, 2, . . . , a − 1, a + 1, . . . , n − 1, C → C + i i+1 i i same color class. As the proof is similar to theorems [3.17, C + C + ... + C , where i = n, n − 1,..., 2 on |P + i−1 i−2 1 3.18] in [12], we omit it. Q| and |P − Q|, we obtain λn−2[λ2 − a(n − a)] and (λ − n−2 2 2) [λ + λ(2n − 4) + 3a(n − a) − 4n + 4]. Definition 33. A double star S{l, m}c is the graph consist- Hence, characteristic polynomial of 2−complement of col- ing of union of two stars K1,l and K1,m together with the ored crown graph is λn−2(λ − 2)n−2[λ2 − a(n − a)][λ2 + line joining their centres. λ(2n − 4) + 3a(n − a) − 4n + 4]. Theorem 34. Let P = {V ,V } be a partition of colored Its color spectrum is 1 2 0 2 a(n − a) −a(n − a) PQ double star S{l, m}c such that hV1i = K1,l and hV2i = , n − 2 n − 2 1 1 1 1 K1,m. Then √ E (S{l, m} )P = (n − 3) + n2 + 2n − 7. where P = (2 − n) + pn2 − 3a(n − a), χ c 2 Q = (2 − n) − pn2 − 3a(n − a) and Proof: Since chromatic number of double star is 2, E ({S0} )P = 2[a(n − a) + n − 2 + pn2 − 3a(n − a)]. adjacency matrix of 2−color complement of double star is χ n c 2 2) Adjacency matrix of 2(i)−color complement of colored O2 C2×l −C2×m 0 P A (S{l, m} )P = C −B J , crown graph is Aχ({Sn}c)2(i) = χ c 2 l×2 l l×m − C J −B m×2 m×l m n×n −Ba −Ja×n−a Ia×b Ja×n−b where B is the adjacency matrix of complete subgraph and − Jn−a×a −Bn−a Jn−a×b In−a×n−b 1 1 ... 1 C = . Ib×a Jb×n−a −Bb −Jb×n−b −1 −1 ... −1 P Jn−b×a In−b×n−a −Jn−b×b −Bn−b |λI − A (S{l, m} ) | = χ c 2 0 P λI2 −C2×l C2×m |λI − Aχ({Sn}c)2(i)| = −Cl×2 (λI + B)l −Jl×m (λI + B J −I −J Cm×2 −Jm×l (λI + B)m Step 1: By using row operations R → R + R and J (λI + B) −J −I 1 1 2 R → R − R , where i = 3, 4, l − 1, l + 1, . . . , m − 1 − I −J (λI + B) J i i i+1 and simplifying further, we get λ(λ − 1)l+m−2 det(D). − J −I J (λI + B) Step 2: On applying the column operation C → C +C + i i i+1 P Q ... + C , where i = 3, 4, . . . , n − 1 on det(D), we obtain Above matrix is of the form . n Q P det(E). Therefore, spectrum is the union of |P + Q| and |P − Q|. Step 3: Then by reducing det(E) along the rows from Then by operating row and column operations on |P + Q| R ,...,R ,R ,...,R , we obtain 3 l−1 l+1 m−1 and |P − Q| as in (1), we get the required characteristic P 0 1 XY 0 Specχ(S{l, m}c)2 = , polynomial of 2(i)−color complement of (Sn)c. 1√ n − 3 1 1 i.e, λn−2(λ − 2)n−2(λ + a − 2)(λ + n − a − 2)[λ2 + nλ − −(n − 3) + n2 + 2n − 7 where X = , 3a(n − a)]. √ 2 Its color spectrum is −(n − 3) − n2 + 2n − 7 Y = and 0 2 2 − a 2 − n + a M N 2 √ , P 2 n − 2 n − 2 1 1 1 1 Eχ(S{l, m}c)2 = (n − 3) + n + 2n − 7. n pn2 + 12a(n − a) where M = − + , 2 2 Theorem 35. Color characteristic polynomial of 2(i)− n pn2 + 12a(n − a) N = − − complement of double star with partition P = {V1,V2}, such 2 2 that hV i = K and hV i = K is (λ − 1)n−4[λ4 + (n − and its color energy is 3n − 2a − 4 + pn2 + 12a(n − a). 1 1,l 2 1,m 4)λ3 + (lm − 2(n − 2))λ2 − 2(lm − 1)λ + (n − 3)]. Corollary 31. Let P = {V ,V } be a partition of (S0) 1 2 n c Proof: Adjacency matrix of 2(i)− complement of col- with vertices of same color class. Then ored double star is O P 0 2 − n 2 −n B C D Specχ{(Sn )c}2 = , 2×2 2×l 2×m n − 1 1 n − 1 1 P Aχ(S{l, m}c) = Cl×2 −Bl×l Ol×m , 0 2(1 − n) 2 2(i) Spec {(SO) }P = D O −B χ n c 2(i) n 1 n − 1 m×2 m×l m×m 0 ... 0 −1 ... −1 and E {(SO) }P = E {(SO) }P = 4(n − 1). where C = and D = . χ n c 2 χ n c 2(i) −1 ... −1 0 ... 0 0 P Proof: Since (S )c is 2−co self and 2(i)−self color |λI − Aχ(S{l, m}c)2(i)| = n complementary with respect to partition of same color class, (λI − B)2×2 −C2×l −D2×m 0 we get color energy of 2 and 2(i) complement of (S )c. −Cl×2 (λI + B)l×l Ol×m n Proof follows from theorems [3.13, 3.14] of [12]. − Dm×2 Om×l (λI + B)m×m Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ Step 1: Reducing the determinant by using row operation Thus by expansion of determinant of order 4, we get 2n−1 Ri = Ri − Ri+1 for i = 3, 4, . . . , l − 1, l + 1, . . . , m − 1 λ(λ − 1) [λ + (2n − 1)] which is the characteristic results in (λ − 1)l+m−2 det(E). polynomial of n complement of colored friendship graph. Step 2: On applying the column operation 0 1 1 − 2n Its colored spectrum is . Ci = Ci + Ci+1 + ... + Cn, where i = 1, 2, . . . , n − 1, we 1 2n − 1 1 l+m−2 P obtain (λ − 1) det(F ). Hence Eχ{(Fn)c}n+1 = 4n − 2 Step 3: On expanding det(F ) along the rows from 3rd row 2) Adjacency matrix of n + 1(i) complement of colored to (l − 1)th row and then from (l + 1)th row till (m − 1)th friendship graph with one of the partite having central vertex l+m−2 row, we get (λ − 1) det(G). and remaining with K2 each is i.e, det(G) = O1 J1×2 ... J1×2 J1×2 λ − 1 + m m − 1 m m J2×1 O2 ... −I2 −I2 λ − 1 + l λ + l l 0 P . . . . . Aχ{(Fn)c} = ...... λ + l λ + l λ + l − 1 0 n+1(i) . . . . m + λ λ + m − 1 λ + m − 1 λ + m − 1 J2×1 −I2 ... O2 −I2 J −I ... −I O Step 4: On expanding det(G), we get required characteristic 2×1 2 2 2 2n+1 polynomial. λ −J ... −J −J l+m−2 4 3 2 i.e, (λ − 1) [λ + (l + m − 2)λ + {(l − 2)m − 2l}λ − −J λI ... I I 2(lm − 1)λ + (l + m − 1)] . |λI − A {(F ) }P | = ...... n−4 4 3 2 χ n c n+1(i) . . . . . i.e, (λ−1) [λ +(n−4)λ +{lm−2(n−2)}λ −2(lm− −J I ... λI I 1)λ + (n − 3)]. −J I ... I λI Step 1: On applying the block row operation Ri → Ri − Definition 36. Friendship graph is constructed by joining P Ri+1, i = 2, 3, . . . , n − 1 on |λI − Aχ{(Fn)c}k(i)|, we get n copies of cycle graph C3 to the common vertex and is (λ − 1)2(n−1) det(D). denoted by Fn. Step 2: By using block operation Ci → Ci + Ci−1 + Theorem 37. If P = {v1, v2, ....vn+1} is the partition of ... + C2, i = n, n − 1,..., 3 on det(D) simplifies to colored friendship graph of order 2n+1 such that central (λ − 1)2(n−1) det(E). vertex is in V1 and hVii = K2 for i = 2, ...n + 1, then Step 3: Expanding det(E) along the block rows starting from 1) E {(F ) }P = 4n − 2. 2nd row till nth row, we get det(F ). χ n c n+1 √ P 2 λ −nJ1×2 2) Eχ{(Fn)c}n+1(i) = 3(n − 1) + n + 6n + 1. i.e., det(F ) = −J (n + λ − 1)I 2×1 2 3×3 Proof: 1) Adjacency matrix of 3 complement of colored By back substitution we obtain the polynomial friendship graph is 2(n−1) 2 (λ − 1) [λ + (n − 1)][λ + (n − 1)λ − 2n]. B2 C2 ... C2 O2×1 P So that Specχ{(Fn)c}n+1(i) = C2 B2 ... C2 O2×1 1 2(n − 1) P ...... Aχ{(Fn)c}n+1 = . . . . . , 1 − n 1 p C2 C2 ... B2 O2×1 −(n − 1) + (n − 1)2 + 8n 1 O1×2 O1×2 ... O1×2 O1 2n+1 p2 −1 1 −(n − 1) − (n − 1)2 + 8n where C = . 1 1 −1 2 √ P P 2 |λI − Aχ{(Fn)c}n+1| = Thus, Eχ{(Fn)c}n+1(i) = 3(n − 1) + n + 6n + 1. λI − B −C ... −C O −C λI − B ... −C O Theorem 38. Let (Fn)c be the colored friendship graph with ...... partition P = {V1,V2,V3} of same color class. Then −C −C ... λI − B O P 1) Eχ{(Fn)c}3 = 4n − 4. O O ... O λ P P 2) Eχ{(Fn)c}3(i) = 4n. Step 1: Expanding |λI − Aχ{(Fn)c}k | along the last row, we get λ det(D) of order 2n. Proof: 1) Adjacency matrix of 3−complement of col- Step 2: By using row operation Ri → Ri + Ri+1, i = ored friendship graph is n 1, 3,..., 2n−1 on det(D), it will reduce to (λ−1) det(E). O1 O1×n O1×n P Step 3: On applying the column operation Ci → Ci − Aχ{(Fn)c}3 = On×1 −Bn Bn O B −B Ci−1, i = 2, 4,..., 2n on det(E) and reducing the matrix n×1 n n 2n+1 along n alternative rows starting from first row of det(E), λ O O we obtain det(F ) of order n. P |λI − Aχ{(Fn)c} | = O λI + B −B P 3 i.e, |λI − Aχ{(Fn)c}n+1| = O −B λI + B The result is got by algorithmic approach. λ + 1 2 ... 2 2 Step 1: We note that 2 λ + 1 ... 2 2 λ(λ − 1)n ...... P λI + B −B |λI − Aχ{(Fn)c} | = λ . 2 2 ... 2 λ + 1 3 −B λI + B Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ 0 −n 2 − n 2 Step 2: Also we can write and its color spectrum is . n 1 1 n − 1 λI + B −B n Thus, E {(F ) }P = 4n − 4. = λ |λI + 2B|. χ n c 2 −B λI + B 2) Adjacency matrix of 2(i) complement of colored friend- Step 3: Applying the row operation R → R − R , i = ship graph is i i i+1 O2 J2 ... J2 J2×1 2, 3, . . . , n−1 and column operation Ci = Ci +Ci−1 +...+ n−1 J2 O2 ... C2 C2×1 C1, i = n, n − 1,..., 2 on |λI + 2B| gives (λ − 2) [λ + 2(n − 1)]. A {(F ) }P = ...... χ n c 2(i) . . . . . Therefore, characteristic polynomial of 3−complement of J C ... O C n+1 n−1 2 2 2 2×1 colored friendship graph is λ (λ − 2) (λ + 2n − 2) J C ... C O 0 2 2 − 2n 1×2 1×2 1×2 1×1 2n+1 and its color spectrum is . n + 1 n − 1 1 Step 1: In |λI − A {(F ) }P |, applying block row opera- Thus, E {(F ) }p = 4n − 4. χ n c 2(i) χ n c 3 tion R → R − R , i = 2, 3, . . . , n − 1 and block column 2) Since chromatic number of friendship graph is 3, adja- i i i+1 operation C → C + C + ... + C , i = n, n − 1,..., 3 cency matrix of 3(i)− complement of friendship graph is i i i −1 2 λ −nJ1×2 0 J ... J J gives [det(λI + C)]n−1 1 1×2 1×2 1×2 −J (λI − (n − 1)C) 2×1 2 3 J2×1 B2 ... −I2 −I2 Step 2: Further simplifying, we obtain J −I ... −I −I P n−1 2 p 2×1 2 2 2 φ {(F ) } = [λ(λ − 2)] (λ − 2n)(λ + 2(n − 1)) and A {(F ) } = χ n c 2(i) χ n c 3(i) . . . . . ...... its color spectrum 0 n − 1 J2×1 −I2 ... B2 −I2 J −I ... −I B √2 n − 1 2×1 2 2 2 2n+1 P Specχ{(Fn)c} = 2n 1 . Step 1: On applying block row operation Ri → Ri − 2(i) √ − 2n 1 Ri+1, i = 2, 3, . . . , n − 1 and block column operation −2(n − 1) 1 Ci → Ci + Ci−1 + ... + C2, i = n, n − 1,..., 3 on √ P P Hence Eχ{(Fn)c} = 4(n − 1) + 2 2n. |λI − Aχ{(Fn)c}3(i)|, we get 2(i) λ −nJ1×2 [det((λ − 1)I − B)]n−1 Observation 40. 1) (Fn)c is 3−co−self and 3(i)−self −J [(λ + n − 1)I − B] 2×1 2 3 color complementary with respect to the partition of Step 2: Further simplification gives the characteristic poly- same color class. n−1 n 2 nomial λ (λ − 2) (λ + n) . P 2) {(Fn)c}3 with respect to the partition of same color Hence its color spectrum is class and {(F ) }P with respect to the partition P = 0 2 −n n c 2 Spec {(F ) }p = and {V ,V } such that central vertex in V and remain- χ n c 3(i) n − 1 n 2 1 2 1 ing vertices in V are non-co spectrum equi-energetic its color energy E {(F ) }p = 4n. 2 χ n c 3(i) graphs. Theorem 39. Let P = {V ,V } be partition of colored 1 2 Conclusion: Generalised color complement of graph not friendship graph such that central vertex in V and other 1 only depends on the partition of vertex set but also depends vertices in V . Then 2 on the assigned colors to the vertices. In this paper, we have P 1) Eχ{(Fn)c}2 = 4n − 4. √ defined and characterised generalised color complement of P 2) Eχ{(Fn)c}2(i) = 4(n − 1) + 2 2n. graph. The color spectrum and color energy of generalised Proof: 1) Adjacency matrix of 2 complement of colored complements of families of graphs are derived. friendship graph is B2 −I2 ... −I2 O2×1 REFERENCES − I2 B2 ... −I2 O2×1 [1] D. B. West, Introduction to Graph theory, Prentice Hall, 2001. P ...... Aχ{(Fn)c}2 = . . . . . [2] F. Harary, Graph Theory, India: Narosa Publishing House, 1989. −I −I ... B O [3] R. Balakrishnan, “The energy of a graph,” Linear Algebra and its 2 2 2 2×1 Applications, vol. 387, pp. 287 - 295, 2004. O O ... O O 1×2 1×2 1×2 1×1 2n+1 [4] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Aca- P demic Press, New York, 1980. |λI − Aχ{(Fn)c} | = 2 [5] G. Indulal, I. Gutman and A. Vijayakumar, “On distance energy of λI − B I ... I O graphs,” MATCH Communications in Mathematical and in Computer I λI − B ... I O Chemistry, vol. 60, pp. 461-472, 2008. ...... [6] Ivan Gutman, “The energy of a graph,” Ber.Math- . . . . . Statist.Sekt.Forschungsz.Graz, vol. 103, pp. 1-22, 1978. I I ... λI − B O [7] Ivan Gutman and Bo Zhou, “Laplacian energy of a graph,” Linear Algebra and its Applications, vol. 414, pp. 29-37, 2006. O O ... O λ [8] N. Biggs, Algebraic graph theory, Cambridge University Press, 2nd P Step 1: We note that |λI − Aχ{(Fn)c}2 | = λ det(C). edition, 1993. [9] R.B. Bapat, Linear Algebra and Linear Models, Hindustan Book Step 2: Employing block row and column operations Ri → Agency, New Delhi, Third edition, 2011. Ri−Ri+1, i = 1, 2, . . . , n−1, Ci → Ci+Ci−1+...+C1, i = [10] E. Sampathkumar and L.Pushpalatha, “Complement of a graph: A n, n − 1,..., 2 on det(C) and further simplification gives us Generalization,” Graphs and Combinatorics, vol. 14, pp. 377-392, the required result. 1998. [11] E. Sampathkumar, L. Pushpalatha, C. V. Venkatachalam and P. G. Therefore, characteristic polynomial of 2−complement of Bhat, “Generalized complements of a graph,” Indian Journal of Pure colored friendship graph is λn(λ + n)(λ + n − 2)(λ − 2)n−1 and Applied Mathematics, vol. 29, pp. 625-639, 1998. Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020) IAENG International Journal of Applied Mathematics, 50:4, IJAM_50_4_01 ______ [12] C. S. Adiga, E. Sampathkumar, M.A. Sriraj and A. S. Shrikanth, “Color Energy of a Graph,” Proceedings of the Jangjeon Mathematical Society, vol. 16, pp. 335-351, 2013. [13] S. D’Souza, H. J. Gowtham and P. G. Bhat, “Energy of generalized complements of a graph,” Engineering Letters, vol. 28, no. 1, pp. 131- 136, 2020. [14] S. Nayak, S. D’ Souza, H. J. Gowtham and P. G. Bhat, “Energy of par- tial complements of a graph,” Proceedings of Jangjeon mathematical society, vol. 22, pp. 369-379, 2019. [15] S. Nayak, S. D’ Souza, H. J. Gowtham and P. G. Bhat, “Distance energy of partial complementary graph,” J. Math. Comput. Sci., vol. 10, pp. 1590-1606, 2020. Swati Nayak received her B. Sc. and M. Sc degrees in Mathematics from Karnataka University, Dharwad, India, in 2009 and 2011, respectively. At present she is working as an Assistant Professor and persuing her Ph.D. under the guidance of Dr. Pradeep G. Bhat at Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal, India. Her research interests include Graph coloring, Graph complements and Spectral graph theory. Sabitha D’Souza received her B. Sc. and M. Sc degrees in Mathe- matics from Mangalore University, Mangalore, India, in 2001 and 2003, respectively. She then received her Ph.D. from Manipal Academy of Higher Education, Manipal, India in 2016. She is currently working as an Assistant Professor Senior Scale at Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal. She serves as a referee for few reputed international journals. Her research interests include Graph coloring, Graph complements and Spectral graph theory. Gowtham H. J. received his B. Sc. Degree from Mangalore University, Mangalore, India, in 2012. He then received his M. Sc. and Ph. D degree from Manipal Academy of Higher Education, Manipal, India in 2014 and 2020, respectively. At present he is working as an Assistant professor at Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal. His research interests include Spectral graph theory and Graph Labelling. Pradeep G. Bhat received his B. Sc. and M. Sc degrees in Mathematics from Karnataka University, Dharawad, India, in 1984 and 1986, respectively. He then received his Ph.D. from Mangalore University, Mangalore, India in 1998. He is currently working as a Professor of Mathematics department, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal. He serves as a referee for several reputed international journals. His research interests include Graph complements, Spectral graph theory and Graph labelling. 1) Modification date: 04-12-2020. 2) We have starred (*) for the 2nd author (Sabitha D’Souza) in the first page below the main title. 3) At the foot note in the first page, we have written *Cor- responding author : SABITHA D’SOUZA, Department of Mathematics, Manipal Institute of Technology, Ma- nipal Academy of Higher Education, Manipal, 576104 India. Volume 50, Issue 4: December 2020 (Revised online publication: 4 December 2020)